Monthly Archives: April 2010

It is the rare adult that intentionally uses strategies to explicitly show respect to others. Our students are learning a lot of things, not just math, and respect has to be one of them.

Last week I wrote about how to teach responsibility. The quick version: give and expect responsibility. This week, respect. The quick version: give and expect respect!

In both cases, it’s harder than it seems, which is my best excuse for writing pages and pages about it. This post will focus on tangible ways to increase the level of respect in your classroom right now. I write as a teacher of high school students at a private boarding school and as a director of a summer camp with 25 staff and 150 campers – two places where respect is vital to my mission. I do not work at a school with a single student who would cuss me out in the middle of class, though I have been called some names and broken up a (singular) fight in the last five years.

Easy Ways to Give Respect

Please say “please” when you ask someone to do something, and thank them after they do it. I direct a summer camp and this is almost a mantra I try to repeat with the counselors. We make little kids do say “please,” right? As adults, we sort of forget to do it, or understand that we’re all comrades and forgo the explicit “please” to… save time? Anyway, saying “please” and “thank you” is probably the easiest way to show your respect for someone. “Take your homework out” becomes “Please take your homework out,” an expectant pause while they do it, and then “Thank you.”

Say “Thank you” when you’re thankful for something. For extra power, name explicitly that for which you’re thankful. “Thanks for being prompt, everyone” or “thanks for getting so into this material, guys, it makes my job really fun!” This has the top-secret side-effect of making it easier to give negative feedback when you need to do that.

Let them know how you’re feeling. You have to trust them before you can do this one, but once you do, it really shows them that, uh, you trust them. “When you guys get into discussing a concept so thoroughly it makes my job fun because the concepts are my favorite part!” If they understand that you have favorite parts, bad days, pet peeves, and personal triumphs (“I’m really proud of this lesson plan”), they will all but have to respect you. You trust them and you are a person and you are telling them all the ways you are trying to help them. How could anyone resist?

Already, with this easy stuff, you are implicitly creating and reinforcing a culture of respect. There’s deeper respect than adding a few words to your diction, though.

Hard Ways to Give Respect

Make the schedule of your class clear, and stick to it. If you say you’ll hand back graded work the next day, please actually do it. If you say there will be a test every Friday, please be ready to give one every Friday, and if you say your office hours will be at 7:30 on Thursdays, don’t change that at the last minute. If you do need to change the schedule, please show respect in that as well, by announcing the change as soon as possible ahead of time. If you change things up all the time, take variable amounts of time to grade work, and are inconsistent with your schedule, please do not expect the students to have their work done when you want it to be, or for them to even show up to class on time!

Find ways to make students part of their own assessment. If you can trust a student to be his own judge (in any part), he will see that you respect him and his time. You’re saying, “hey, I’m getting paid, and you’re required to be here by law. I think you should have some say in what goes on here.” If you can find a way to let students guide the class as a whole, that’s even better. “You’ll go to jail if you leave (or whatever happens), but while you’re here, I want to do my best to make this interesting and enriching from your point of view.”

Tell students what you’re trying to do with the class. Just tell them everything. This was hard for me in some sort of ego way I can’t describe.

Apologize when you waste a student’s time. The obvious corollary is Try Not To Waste Students’ Time, but you’ll fail in that for at least some kids (sorry, I’m a pessimist I guess). When you do give a stupid assignment or are unprepared for a lesson plan (not that I’ve ever been unprepared!) just apologize for it.

State the objectives of your lesson. It’s OK if they know your plan. Bill Ferriter, a NC county teacher of the year, explains why the pedagogy is good. But it also just shows your students that you want them involved in the process. You think they’re smart enough to be in the know.

State everything else. “I was hoping you would do xyz.” “I never thought of that!” “It’s frustrating to me that you won’t focus on this.” “Today we’re going to try something that I’m a little worried about” (and then all the reasons you think it’ll be great, of course).

These harder things to do are hard for me because I can’t remember to do them all the time. Some of them are a lot of work and some of them just don’t occur to me naturally. Why do the students need to know what’s going to happen – they’re about to find out! But intentionally striving to express your respect is worth some extra work to me, and, I’m not kidding you guys, I’ve seen these methods improve my classroom and camp culture dramatically. Please try these out!

Explicitly Appreciate Respect You Receive

All the same reasons, and all the same benefits. “I appreciate you letting me know that you’ll have to miss a class next week,” “thanks for getting back to me about this,” “thanks for waiting so patiently while I answered Sarah’s question,” “it was so nice of you to think of this!”

Don’t Accept Disrespect

Frankly, I have much less experience with this than some other teachers at my school. I don’t know if my methods of fostering respect are just SO GOOD, or that I mostly teach older students, or what, but I don’t get a lot of disrespect directly. More often one student will disrespect another in some way. Whether I get it or a student gets it, I address it directly and immediately by saying briefly something like “that was disrespectful, and disrespect has no place in our school.” I’m serious when I say this. I’ve got a controlled anger in my voice even for such a small infraction as “shut up” (“shut up” usually gets said exactly once per year in each of my courses). I ask the students involved to (please) find a respectful way to express whatever they want to communicate.

This is a high priority for me and I will stop a lesson to talk about respect even when we’re a week behind my original schedule (“sorry class, we have to change the schedule because I made unrealistic estimates/you guys aren’t in to it/you guys are so into it”).

I hope you’ll give some of this a try. I bet all of you think of your students with respect, and I’m sure almost all of you treat your students with respect. It is the rare adult, though, that intentionally uses strategies to explicitly show respect to others. Our students are learning a lot of things, not just math, and respect has to be one of them. Please audit your own communication methods and see if there are any ways you can build more respect into them!

My life is changing in many major ways. I just got engaged. I just bought a house with my fiancé(e?). I’m getting ready for the summer as a camp director, and preparing to start my own software company (seriously!). Tomorrow I leave on a 7-day hiking trip in Missouri with 8 students, and won’t be able to check on my house, my staff, or work on the science fiction class I’m teaching in 10 days, and I’m really not ready to go.

Despite the toll these changes are taking on my time, I’m happy to host the Math Teachers at Play blog carnival, edition #25. More and more, the web is becoming a tool which people use to learn, teach, and learn by teaching (and teach by learning!). Much of my own expertise as a teacher, as a programmer, and as a camp director has come from the internet, and my journey into the blogosphere this year has given me a chance to give back (and to grow from that process). This blog carnival was one of the first hubs of interconnectivity I stumbled upon in my search for quality contributors to said blogosphere 1.

I also want to take this opportunity to share all of the files I’ve generated this year. I’m proud of them as a record of my progress, but you’ll be disappointed if you expect a continuous style or theme. You’ll see my philosophy change from unit tests to SBG tests, you’ll notice when I got my CPM books, and when I ran out of steam for my notes templates and slide decks. Please feel free to use, alter, and redistribute any of my work. My ego says “and link back to me here” but I don’t think I actually even care about that. I hope you can use them! Sorry: they’re all in Office 2007 formats.

And now, with all of my sentimental blathering aside, the Carnival itself! The theme this month is sharing connections. I’ve arranged every blog submitted according to the number of subscribers that google reader tells me it has. Those without a lot of connections are listed first, so veteran bloggers can take a while to check them out and leave some comments. Those with many connections are listed later, so that newcomers to blogging can start with some popular blogs with already-active comment sections. If you’d like to submit your own article to next month’s carnival (hosted at math hombre), make sure you do so at blogcarnival.com before May 20!

Thanks for keeping this community strong by writing, reading, and commenting on all of the great math ed blogs out there. It’s really something – and what a great way to set an example of life-long learning for our students!

Riley

I can’t believe how comfortable I’ve gotten with these ridiculous words ↩

Graceachenrecently wrote a post that got me thinking about ways to teach the “stuff” that is hard to assess accurately and even harder to teach explicitly. For example, I would like to teach all of my students:

Responsibility

Respect

Curiosity

Investigative skills

Teamwork skills

To be comfortable with a lack of knowledge and with mistakes

But I’m not going to make a standard for each of these things and assign grades, and no section of any one of my lesson plans will start, “here’s one way you can be curious1.” However, I do teach these things intentionally and I do assess them. In this post, I write about how to teach responsibility, and about some traps that seem important to avoid. There are two types of responsibility: the fulfill-your-obligations kind and the take-ownership-of-your-destiny kind. I’m talking about both.

Find Ways to Give Ownership, and Communicate Them

Give students the answers to their homework

Matt Townsley and grace said that they use homework to teach responsibility. I do this too and so do many other teachers. The three of us make sure that students have answer resources available before assignments are due so that they can more easily feel responsible for the quality of their work. Without answers, if a student does his work incorrectly, the excuse is easy: “I didn’t know how,” or “I guess there’s a mistake.” With solutions, a student is denied these reasons for poor work.

Tell students, “the solutions to these problems are available to you so that you can make sure you understand; please make sure you understand.” This communicates that it is their responsibility, that you think they can do it, and that you trust them to do it.

Refer to student work and expertise during class

Please have students create projects that explain or summarize the concepts you’re studying. When you’re lecturing, or when a student has a question, refer to those projects. Whomever made that project will be directly responsible for the knowledge being disseminated. You might also make different groups of students “expert groups” in different areas, and ask those groups specific questions (but please do give them the tools to actually become experts before putting them on the spot).

Again, it’s not always easy to think of telling the students that you’re doing this, but saying “these posters will be used as reference for the rest of the year” will let the kids know that you expect them to make production-quality work. Actually using them will show them that you expect them to make production-quality work. “These posters are going live, kids.” They won’t mess it up after you’ve articulated (and demonstrated) such high expectations of them.

Use Standards-Based Grading (if you must grade)

Or at least, some grading system as clear. Please don’t use a grading system that obscures the reasons for your students’ grades (like averaging), because then the focus will shift from the responsibility of the knowledge to the responsibility of the grade. I think grades should be as invisible as possible. SBG is the best grading system I’ve seen because it gives the clearest idea about what a student needs to learn to improve his status.

You can only hold a student responsible for his grade after convincing him he can control it, and showing him how the controls work. If, at any point, he loses that control, you won’t be able to expect anything more out of him.

Model responsibility (visibly)

Let your students see something you’re working on. Let them in on your process for their class! Show them all the ways you practice, prepare, and follow up. Some of my students read my blog (hey guys!) and they see me in their other classes, observing other teachers. Unique to Scattergood, perhaps: they see me attending required community events, keeping my apartment clean, helping with dinner cleanup when it’s my turn.

Don’t take responsibility away

Don’t grade homework

You shouldn’t grade homework because doing so transfers responsibility in a bad way. When students have to do homework for their own sake, they are being responsible for their own knowledge and edification. When you give them a grade, they will change – they will think themselves responsible only for that grade. Please also tell your students why you are not grading homework – the effect doubles when they know what you’re trying to do.

There are many arguments for grading homework, but I haven’t heard one yet that’s convinced me. Let’s debate in the comments if you’ve heard one that’s convinced you.

Avoid assigning unnecessary work

No one will feel much passion for work that they consider pointless. If you really think that Johnny needs to do 30 factoring problems (lots of good arguments for assignments like these), then please explain to Johnny why you think so. It may take something more convincing than mere explanation.

If a student thinks your whole class is unnecessary work, you have a more fundamental problem on your hands, and you might consider starting there.

Of all of my suggestions, which are all hard to do, I think this is the hardest. If you’ve got a bunch of kids at different levels, 140 hours, and 140 skills to teach, you may have to give assignments that are not individually tailored to each student’s level – but you can at least tell them that!

Don’t answer questions that your students have the tools to answer

In my experience, I am my students’ favorite resource. They will ask me a question before they use any other resource, including their notes, their books, or even their own understanding. If you think that a student should be expected to be able to answer a certain kind of question, expect it of her! Please do not waste your own effort by completing parts of your student’s lesson for her! Don’t be too helpful!

None of these are explicit ways of teaching responsibility. You make them explicit by talking about them, but the real lesson lies in their practice. The real lesson happens when a student didn’t do some homework and can connect that with the fact that he’s feeling left out in class, or when he doesn’t understand something he’s expected to. The real lesson happens when his classmates are looking at the crappy poster he made and complaining to him about how inaccurate or incomplete it is, or, better, when his classmates are looking at the awesome poster he made and complimenting him or incorporating it in their own work. I guess that for my suggestions to work, your classroom culture needs to be set up in a way that allows these feelings to happen, and you need to be on the lookout for them, ready to emphasize them and focus them.

Assessing responsibility is more vague in my mind than teaching it. I mean, ultimately, if a student passes a class, she’s responsible for that, and if she doesn’t, she’s responsible for that too. If you get the opportunity to give written feedback to your students, or to have one-on-one meetings with them, make sure to emphasize what you saw working and not working towards the goal of passing. Any other ideas out there?

This is a fascinating topic to me as a teacher and as a member of society contemplating parenthood. Expect more, focusing on my other bullet points!

We were reviewing today. We spent 15 minutes brainstorming for topics we’ve studied this year. Students called topics out and I wrote them on the board.

Wouldn’t it have been clever to have a student write them on the board? I mean, why not? Three students could have done five minutes each, or something. I could have participated in the discussion just as much, but I would have imparted some power (and the complementary responsibility) to students so easily.

Once you start thinking about ways to give students more power/responsibility, you see them everywhere. What have you done to change the balance of power in your classroom – either towards the students or towards the teacher?

For several years I have used the frequencies of sounds to give students practice using exponents. This year I did things backwards: we’ve been studying exponents in various abstract ways first, and then as a sort of conclusion / practice / experiment / “hands-on” kind of thing brought out the tuner.

5 minutes: have a student read the first part of the first question aloud. There is a lot to read here – too much to give to each group to read, I think. After “the chromatic scale,” have groups turn on g-tune and attempt a chromatic scale. This will be fun and hilarious for them and perhaps more so for you. Make sure each group has g-tune working and can identify the notes that it is displaying. Surprisingly, many groups could do this simultaneously – my groups were less than 10 feet apart and did not suffer too badly from interference from nearby singing.

10 minutes: have a student read the next part of the first question. Stop after “number to the note,” or whatever you have rewritten it to be (please do rewrite it). Explain that your goal for the class period is that the students will be able to find a mathematical pattern in the musical scale that will allow them to predict the frequencies of different notes (and the notes at different frequencies if you’re studying logs too). Then let each group read the discussion questions aloud and decide on their numbering system. Today I had groups start with 0 at C4 and C3. Another group started with 0 at G#5 and go backwards, and another group started with 1 at A2 and go up by 0.1 (so C3 was 1.3). Of course the scale does not matter as long as it’s linear, so whatever will be easiest for them to work with will be best. Probably the easiest scale would go up by twelfths, but of course you will not tell them that.

10 minutes: students will find the specific frequency of ten different notes. After they have done this and recorded their data in duplicate or triplicate, you (the teacher) can take their sheets and swap them between groups to increase measurement speed. You are now done with my worksheet, and luckily have added on to it the last half of this lesson, which I did not have time to do before class.

10 minutes: Ask groups, “do you see any patterns in the data?” Your rewritten worksheet says something like “look for patterns in the data.” If they don’t see any, ask how they can look for patterns in numbers. They should be thinking of strategies like “make a table,” or “make a graph.” You might be extra-direct and prompt them to look for a relationship between C3, C4, and C5, or F3, F4, and F5, etc. Today in my class 100% of students noticed that the frequencies approximately double between notes one octave apart.

10 minutes: Ultimately you want to be able to bring this back to exponents and logs. I had students graph their data on geogebra, and I asked questions like “what would the frequency for C6 be?” and “what note would be at 1000 Hz?” Inevitably their answers included the exponents and logs (though only one group called their logarithms logarithms).

If you are not a small-group kind of teacher, the closure of this lesson seems weak. But just wait until you hear the kind of discussion that happens with 3 or 4 kids trying to figure this out, with such a fun activity (singing) and fun tool (cool waves and stuff that respond to your voice) and high skill levels (my kids can already work with exponents and logs relatively comfortably). One of my four groups actually found the equation that best matched their data, AND its inverse – it was really neat to see it in geogebra. Every group used exponents to talk about the frequencies, and all but one group started to talk about logarithms too. This is practice using math in a casual way and feeling the fun and power of it.

This is a fun class full of joyous noise, and the kids were really into measuring precisely and graphing precisely. Every group eventually made a graph and noticed it looked exponential or logarithmic (depending on what they assigned to which axis). They got to think about what kind of scale makes sense in the first question, and got to see exponents at work in nature. I used the last five minutes of class to indulge in a monologue about the philosophy of sound: “is ‘sound’ the only way to interpret vibrations?” and “the computer registers 180 and we hear a certain pitch – which is more useful?” etc.

Sometimes when I am self-deprecatory I am actually trying to emphasize how awesome I am. In the case of this worksheet, however, please be advised that you probably actually want a much better one. For one thing, the worksheet provided only covers half of the lesson plan (ran out of time!). ↩

My blog has gotten a little lofty lately, and it’s been a while since I just posted some plain old good ideas you could use tomorrow. Here’s one if you have access to a class set of laptops or a computer lab: have your students make tools in geogebra. I’m not going to try to frame this in a lesson plan – it’s just a tutorial for you. Open geogebra to follow along.

We’ll make a midpoint tool1 in two different ways today. The first way will be geometrically, via construction. The second way will use an algebraic formula. This might be a fun way to connect geometry and algebra! If you already know both of these methods, skip down to the “Toolify” section. If you already know how to make tools, skip down to “The Point” section at the bottom!

Midpoint via Construction

To make tools in geogebra, you first do what you want the tool to do, and then tell geogebra about it. So to make our midpoint tool via construction, we first have to actually do the construction.

Start with two points.

Use the circle tool to draw a circle with center A and perimeter point B. Draw a second one with center B and perimeter point A.

The circles intersect at two points. Use the line tool to draw the line between them. Also draw the line from A to B.

Of course, these two lines intersect at the midpoint between A and B. Use the point tool to give it its own name.

A crucial step: test your construction by moving the points A and B. The entire construction should move, but E should still be the midpoint. Do not move on if your construction does not pass this “wiggle test.”When your construction passes the wiggle test, go to the “Toolifying” section below.

Midpoint via Algebra

We’ll do this construction entirely from the input bar. Text in bold is text you can type directly into the input bar.

A = (2, 4)
B = (5, 6)Typing these commands creates two points, A and B, at the specified coordinates.

x_A = x(A)
y_A = y(A)
x_B = x(B)
y_B = y(B)These commands create variables with which you can access the coordinates of points A and B. The thing on the left of the equal sign is the NAME of the variable. The thing on the right of the equal sign is the VALUE of the variable.

Do the wiggle test on your variables. When you wiggle points A and B, all four of the variables from step 2 should change. You can move point A by dragging it with the mouse, or by redefining it with something like A = (1, 4). Do not move on until your variables have passed the wiggle test.

When this point E passes the wiggle test, move on to “Toolifying” below.

Toolifying

Regardless of HOW your construction was made – via algebra, geometry, or even calculus2 – if it passes the wiggle test, you can make it into a tool.

From the “Tools” menu, choose “Create New Tool.” You’ll be presented with a dialog like the one below.

The most crucial part here is to identify to geogebra your output object. What is the RESULT of your tool supposed to be? In our case, we were trying to make a tool that finds the midpoint of two points. The output is that midpoint. We called it point E. Select that object from the list. You could also click on that object from the graph view.

Go to the “Input Objects” tab. On this tab you will select the objects that your tool needs to work. Our tool is supposed to create the midpoint from two starting points, so those two starting points must be listed as input objects.
So far, in my experience, Geogebra has always guessed the necessary input objects for me. Point A and B are already listed because geogebra knows that they are at the root of your construction. This will save a lot of confusion with your students.

Head to the “Name & Icon” tab to personalize your tool. The “Tool Name” is what will appear on the tool bar. The “Command name” is what you would type on the input bar to use your command. The “Tool help” will appear in the toolbar when your tool is selected.

After you click Finish, your tool is created and ready to use. Let’s test it! First, make two new points.

Then, choose your tool from the toolbar and click those two points, one after the other. The order you click is important in some tools: the first object you click becomes the first input object, the second you click becomes the second input object, etc. If everything is working, you should see the midpoint appear between your two new input points!Remember to try the wiggle test by pressing escape and dragging F and G around! You will not be able to drag H around – geogebra cannot (yet?) run tools backwards like that. Note that geogebra does not create all the intermediate objects needed for the construction. If you want those objects to appear, include them in the list of output objects when you create the tool.

Your students will enjoy this and feel a sense of ownership of the math. It’s fun to use the tool on its own output, for multiple nested midpoints and things like that. You can even create other objects with the output of your tools for extra fun. Below is an actual geogebra applet – drag the blue points around for fun!

This post is about some of the virtues of programming computers in math class. I include a long anecdote and a quick geogebra tutorial. The punchline: teaching kids to program introduces them to an environment that gives instantaneous, continuous, 100% correct, 0% helpful feedback without judgement. The computer doesn’t say, “you’ve made a mistake here,” it just shows you a result, and it’s up to you to interpret it, decide if it’s a correct result, and find the problem if it’s not.

In my calculus class we’re looking for a way to guess how long it will take a glass of cold water to rise to within a few degrees of room temperature. We’ve taken a lot of data, and discovered that 1. However, no one in class could find a that satisfied the equation (not even ). So I broke it down a little:

If the water is 40 degrees right now, what do you think its temperature will be in 1 minute?

You can imagine where it went from here – lots of guesses, including some really good ones and pretty bad ones. Instead of helping them write it in math (I didn’t even tell them that this is, like, a method), I took a little time to teach them some geogebra 2 programming techniques so they could flesh out the ideas themselves.

We started with a blank file and created a point, A. I showed them how they could make a point that was 1 unit right of and 1 unit above A. Type ( x(A)+1, y(A) + 1) into the input field below to get a taste of this.

They liked this; I’m always surprised by how much students like stuff like this. So this alone has them interested in learning geogebra. Then I lay the geogebra Tool Manager on them.

In the applet below, open the “Tools” menu and choose “Create New Tool…” and choose B as an output object. You’ll notice that A has been chosen automatically as an input object. Choose the name and icon you like from the third tab.

What follows is a picture to help you – it’s not an actual geogebra applet! Don’t try to click on it!

helpful image

After you’ve gone through all the tabs and clicked “Finish,” your new tool will appear in the toolbar, all the way to the right. Now you have a tool that accepts a point as input and creates a point 1 to the right of it and 1 above it, automatically! Try clicking around with your tool. You can even click on the output of the tool to feed it back into the tool, creating a long line.

Well, at this point my students were ready to get back to the temperature thing (we’ve been working on it for maybe 3 hours at this point, over several classes). With a little nudging from me they make a tool that does Euler’s method (fixed width of 1) on a point, by choosing B = (x(A)+1, y(A)+0.01(70-y(A))) and using the tool repeatedly. Check this out – try dragging the point A up and down to different temperatures!

Note that now I can ask them questions like “how do you make the curve flat?” and “why does the curve change direction?”

With a few simple commands the students have created a complex piece of software, and they can be proud of it. It has a pleasing organic flow that they like intrinsically. But the best reason to teach them to program geogebra is that geogebra programs only work if they are mathematically sound. I can see in an instant whether a student has created the program correctly or not. When they go on to create other geogebra programs, I can assess whether they understand the concept or not, and more importantly, they can assess their own knowledge. Geogebra will show students if they understand or not, but won’t give suggestions or hints. It also doesn’t mind if they are wrong 400 times in a row.

I start teaching geogebra commands to kids right away, in the first month of school. My algebra 2 kids can transform an arbitrary function, calculate a line or other curve through 2 points, and animate sliders. The ones that get really interested learn more on their own (one student has all but mastered latex). The kinds of assignments I can give and the kinds of exploration they can do now are really something else. Please teach your students to use computer technology well. It’s not enough that they can use the built in functionality – they have to be able to make their own.

The kids came up with many ideas about temperature change and I directed them towards this equation ↩

If you’re a math teacher and you don’t know how to program geogebra, I recommend looking into it – it’s fun and extremely helpful for creating interactive diagrams. ↩